infrared and raman selection rules for lattice vibrations: the correlation method

FEATURE

Infrared and Raman Selection Rules for Lattice Vibrations: The Correlation Method

W. G. Fateiey

Chemistry Department, Mellon Institute of Science, Carnegie-Mellon University, Pittsburgh, Pennsyl- vania 15213

Neil T. McDevitt and Freeman F. Bentley

Air Force Materials Laboratory (LPA), Wright-Patterson Air Force Base, Ohio 45433 (Received 21 December 1970) INDEX .HEADINGS: Selectkm rules for lattice vibrations; Correlation method.

ARTICLE

INTRODUCTION

With the recent growth in interest in the ir and Raman spectra of crystals, it has become very impor tant to know which vibrational modes are optically active. Hornig, ~ Winston and Halford, 2 and Bhagavan- tam and Venkatarayudu a pioneered in developing methods for this. However, heretofore the determina- tion has been a laborious procedure fraught with difficulty and with many points of indecision. Among the lat ter is the choice of the primitive cell and the correct site symmet ry of each atom. What is needed is a short, straightforward, foolproof method. We propose here an al ternate procedure for obtaining the act ivi ty of the vibrations from the correlation tables which comes close to meeting these goals. The calculation is reduced to but a few minutes' work. The method will be explained in detail by use of numerous examples.

The correlation procedure has been discussed previously in several papers and books dealing with lattice vibrations. 2-5 We have chosen not to review the theory, but to proceed directly to a demonstrat ion of how to use it and the correlation tables to obtain the vibrational selection rules for solids.

I. RULES

An orderly procedure is outlined below for the stepwise calculation of selection rules predicting ir and Raman activity. The reader is warned that in some cases there may be slight variations in some of the steps used in this method. However, intelligent reason- ing will help him through these cases.

A. Crystal Structure

The crystal s t ructure of the sample must be known. Alternatively one can be assumed, predictions made for the vibrations, and then these predictions can be compared with observations to prove or disprove the

assumed structure. I t is far better to know the structure in advance, however.

The crystallographic information may be obtained from Refs. 7 and 8 or from the original l i terature. Examples of tha t which is needed are given in Table I. 6

B. Molecules per Bravais Space Cell

The Bravais space cell is used by molecular spectros- copists to obtain the irreducible representat ion for the lattice vibrations. The crystallographic unit cell may be iden t i ca lwi th the Bravais cell, or it may be larger by some simple multiple. This can be ascer- tained from the capital letter in the x-ray symbol for the crystal structure. For all those crystal structures designated by a symbol containing P (primitive), the crystallographic unit cell and the Bravais unit cell are identical. (An example from Table I is Pun,, for SrTiOa.) Crystal s t ructure designated with other capital letters (B, C, I , etc.) have crystallographic unit cells which contain two, three, or four Bravais cells. (An example from Table I is I41/ama for TiO>) The irreducible representations obtained from these crystallographic unit cells will contain two, three, or four times as many vibrations as are needed to represent the lattice vibrations of the crystal. This problem of including too many Bravais cells in the crystallographic cell can be eliminated by dividing the number of molecules per unit crystallographic cell by a small integer. This integer is identical with the number of lattice points (LP) in a crystallographic cell of specific symmet ry as designated by the capital letter in its symbol. Table I I gives this number, LP, which reduces the size of the crystallographic unit cell to the desired Bravais space cell. This reduction has been included in Table I.

In summary, the number of molecules in the Bravais space cell = Z / L P = t h e number of molecules in crystallographic unit cel l /number of lattice points Z B (from Table II) .

Volume 25, Number 2, 1971 APPLIED SPECTROSCOPY 155

Table I. Cyrstallographic information for several examples.

C r y s t a l s t r u c t u r e n o m e n c l a t u r e ~

C r y s t a l X - r a y S p e c t r o s c o p i c

L a t t i c e M o l e c u l e s M o l e c u l e s pe r p o i n t s b p e r B r a v a i s u n i t cel l (Z)~ ( L P ) cell ( Z / L P ) - Z ~

SrTiO3 Pm3,~ Oh 1 TiO2 ( a n a t a s e ) I41/~,.,~ D4h 19 ZrO2 P ~ / . Cab 5 ~-Al~O3 RS~ Dad 6 Cu~O P.3,,, Oh 4 N H 4 I ~ P4/ . . . . . D4h 7

1 1 1 4 2 2 4 1 4 '2 1 "2 '2 1 2 2 1 '2

See Refs. 7 and 8. b See Ref. 5. e Phase I I I (see Ref. 6).

C. Site Symmetry of Each Atom in the Bravais Cell

The equilibrium position of each a tom lies on a site which has its own symmet ry . This site s y m m e t r y is a subgroup of 5he full s y m m e t r y of the Bravais unit

• cell. I t is ve ry impor tan t to ascertain the correct site s y m m e t r y for each a tom. I t is easy to do this in some cases, difficult in others. Let us consider the following examples.

1. Cu~O

Table I s tates tha t the s y m m e t r y is Oh 4 and tha t there are two Cu20 units in a Bravais cell. There are, therefore, four equivalent copper a toms and two equivalent oxygen a toms in the Bravais unit cell. (This rule is always applicable provided one has found the equivalent a toms in a crystallographic cell. This information is provided with the crystallographic s t ructure in Refs. 7 and 8. Appendix I contains this information.) Next, one turns to the table in Appendix I and looks for the ent ry Oh 4 in the third column. I t is No. 224. In the r ight-hand column are t abu la ted all the possible site symmetr ies for this space group. These are wri t ten as T~(2); 2D3d(4); D ~ ( 6 ) ; . . - . They are given in full in Table I I I . These are all the possible kinds of sites for an Oh 4 crystal, but most of them will not be occupied in a specific crystal•

The most useful informat ion is the number contained in parentheses, for it is the number of equivalent a toms which have tha t par t icular site symmet ry . For example T~ (2) means tha t there are two equivalent

Table II. The number (LP) which reduces the crystallographic unit cell to the Bravais space cell.

T y p e of c r y s t a l s t r u c t m ' e No. ( L P )

A 2 B 2 c 2 F 4 I 2 P 1 R 3 or 1 ~

Here the crystallographic group may have already been decreased by three; if so, one need n o t divide the crystallographic cell. Simple indication of whether one should divide by three or n o t is found in the example of ~-Al~O3 (Table I) which contains two molecules per unit cell. Certainly, one would not divide by three in this case.

atoms occupying sites of s y m m e t r y Ta. Similarly Dan(4) indicates the presence of four equivalent a toms on Dad sites.

Some of the site symmetr ies have numerical coefficients, such as 2D3j (4) in Table I I I . The coefficient 2 means tha t there are two different and distinct kinds of D3~ sites in this unit cell. Each kind can accommodate four equivalent a toms. In a given crystal there m a y be a toms on one or both of these sites, or on neither.

The second and third columns of Table I I I i l lustrate the above remarks.

For Cu20 the x-ray results show tha t there are four equivalent copper a toms and two equivalent oxygen atoms. Wha t will be their site symmetr ies? F rom Table I I I (or f rom the equivalent en t ry in Appendix I ) one sees tha t there is only one site s y m m e t r y which can accommodate four equivalent a toms : D:~,~. There- fore, the site s y m m e t r y for copper is D:~j. Similarly only one kind of site can accommodate just two equivalent a toms : T~. This therefore is the site sym- me t ry for oxygen.

In selecting the site symmet ry , one mus t always have the number of equivalent a toms equal to the accommodat ional value of the site symmet ry .

The above example was a typical ly simple in tha t there was no ambigui ty in the result. We turn now to another example which is slightly more difficult.

2. TiO=, (Anatase Form)

Table I tells us tha t the space group is deuignated D4h '9 o r I q / a m d , with two molecules per Bravais unit cell. There are therefore two equivalent t i t an ium a toms and four equivalent oxygen a toms in this Bravais cell. F rom Appendix I we find tha t this is No. 141 and has the site symmetr ies 2D2d(2); 2C2,,(4): C.a,(4); 2C2(8) ; C.,(8), C1(16).

Consider first the two equivalent t i tanium at, oms. Only the D2~ sites accommoda te two a toms ; therefore, it follows directly t ha t the t i t an ium a toms are on sites of D2d symmet ry . There are two separa te kinds of such sites (coefficient 2), but it not necessary for us to know which kind is occupied.

For the four equivalent oxygen a toms there are two possible site symmet r ies : C2h and C2~. Both w i l l

156 Volume 25, Number 2, 1971

Table III. Site symmetries for the space group designated O~, 4 or P.~ or 224. (Cu~O).

No. of equivalent atoms on this No. of kinds of site of Bravais sites of this

Bravais cell cell (No. in symmetry site symmetry parentheses) (coefficient)

Td (2) 2 1 2D3d (4) 4 2 D2a(6) 6 1 C3, (8) 8 1 D,(12) 12 1 Csv (12) 12 1 3C2(24) 24 3 C,(24) 24 1 C1 (48) 48 1

accommodate four equivalent atoms. One or the other will be correct, but addit ional information is needed to make the choice. For this one turns again to the crystal lographic tables, 7,s which s ta te tha t the oxygen a toms lie on C2~ sites. (See Appendix I I for an explana- t ion of how to use the crystallographic tables.) This addit ional information is needed whenever one meets an ambigui ty of the above type.

Table IV gives the site s y m m e t r y of each a tom in the various examples used in this paper.

D. C o r r e l a t i o n of the Site G r o u p to the Factor Group

Different authors va ry in their choice of the t e rm factor group, crystal group, or correlation group to describe the crystal symmet ry . All those te rms are equivalent.

Previously the site s y m m e t r y has been found for each a tom in the lattice and these results are summarized in Table IV. Now the s y m m e t r y species will be identified for each equivalent set of a tom ' s dis- p lacements in tim site. These displacements we describe will become the lattice vibrat ions in the crystal. Knowing the site species for these displacements, the correlation tables relate each species of site groups to a species of the factor group. This correlation explicitly identifies the species of the latt ice v ibra t ion in the crystal and fur ther allows prediction of ir or R a m a n act ivi ty. Using the following molecules as examples, first the lattice modes will be identified in the crystal by obtaining the irreducible representat ion which contains the number and species of the lattice vibrations and, second, the ir and R a m a n act iv i ty of each vibrat ion will be described.

1. TiO~ Crystal

As summarized in Table IV, the two t i t an ium atoms are in D2d sites and the four oxygen a toms are in C2~ sites. In this example, each set of equivalent a toms will be t rea ted separately.

Ti tan ium Atoms. First, the vibrat ional displacements of these t i t an ium a toms in the lattice can be described as simple motions parallel to tbe x, y, or z

Table IV. Site symmetry of each atom in various examples.

Ti02, Example Anatase SrTiO:~, Cu~O A1203, ZrO2, NHd

Site of Ti-D2d S~0h Cu-D.~a A1-C3 Zr-C1 NH4-D2a equivalent O-C~ Ti-Oh O-T,i O - C 2 O-C1 I-C4~ atoms O-D4h

axes. The simplified description of the vibrat ional mode allows easy classification into one of the species of the site symmet ry , D2~. For example, the displacements of the t i tan ium atoms parallel to the z axis will have the same character as the t ranslat ion in the z direction. The t ransla t ion Tz belongs to the species B2 of the site group. Therefore the a tom displacements parallel to the z axis will also belong to the B2 species. Similarly, the displacements of t i t an ium atoms along the x axis will have the same character as T , and belong to species E. I t is impor tan t to note here tha t this approach, which classifies the lattice v ibra t ion as excursions in x, y, and z directions, is no different f rom the descriptions used for molecular vibrat ions such as bond stretching, bendings, and twistings. Of course the normal vibrat ions in a crystal or a molecule are far more complex than this simple picture provides; however, the impor tance of this method is found in the simplicity with which the lattice vibrat ions can be classified.

When the species of the site group is identified for each lattice vibration, then this information is integrated via the correlation tables to the species of the crystal which contain this lattice vibration. To begin this correlation procedure, Table V gives a port ion of the D2~ site group and identifies the species of the t ranslat ions T~, T~, and T , (See Ref. 9 for character tables.) Since the lattice vibrat ions have the same character as the translations, the species which contain these vibrat ions can readily be identified and this information is presented in Table V.

Before applying the correlations of site to factor group, it is convenient to define some useful t e rm to aid in this integration.

(1) t ~= The number of t ranslat ions in a site species ~. This number can take the values of zero, one, two, or three depending on whether none, one, two, or three translat ions are contained in the site species % respectively. This information is readily available from the character table (see Ref. 9).

Table V. Species of the site group D~ and the translations.

D~d Site of Ti Translation Ti Atoms atom, species species excursions

Ai A~ B1 B2

E

T, Motions parallel Io z axis

T,. u Motions parallel to x and y axes

APPLIED SPECTROSCOPY 157

R ~ = T h e number of rotations included in the site species ~. Again this value will be either zero, one, two, or three. The character tables in Ref. 9 clearly identify the rotations as Rx, R~, and R,.

(2) i f = The degrees of vibrational freedom present in each site species 7. This can be calculated as follows, where n = n u m b e r of atoms (ions or molecules) in an equivalent set

t ' . n = f ' . (1)

f R ~ = T h e degrees of rotat ional freedom present in each species y. This can be calculated by modifying Eq. (1) to give

. f ~ = R ~. n. ( l a )

(3) a~ will represent the degrees of freedom con- t r ibuted by each site species ~ to a factor group species ~'. The value of a~ can be calculated as follows:

f f = a r e ~Cr. (2) r

The derivation of this equat ion will not be presented here; however, as one understands and uses this method, the origin of this equat ion will be obvious. The values of Cr are given below.

(4) Cr is the degeneracy of the species f of the factor group. An additional superscript, % may sometimes be added to show its correlation to a species of the site group. Usual values of Cr are summarized below.

Species Value of Cr

A 1 B 1 E 2 F 3 G 4~ H 5

Usually the species designation has either a superscript, e.g., A', A", etc., or a subscript, e.g., A~g, E, F ~ , etc.; however these super- and subscripts in no way describe the degeneracy of the species and for this reason are not included herein.

There are exceptions to the above description of degeneracy for certain correlations where separable degeneracy exists. Without a proof, the following modification of the existing correlation tables will give the correct correlation and a~ values.

(1) Point group C6, C4h, C6h, $6, T, and Th do not use the 2 coefficient which appears in these correlation tables for the doubly degenerate, E~, species.

(2) For a portion of the T and T~, point group correlation tables, modify as follows.

T C3 Th $6 C~

F A + 2 E Fg Ao+2Eg A + 2 E F~ A,~ + 2Eu A + 2E

Here, a coefficient of 2 is added to the E species which must be doubled because of the separable degeneracy.

(5) Convenient checks. I t is helpful to check the bookkeeping as the correlation method progresses. The following equations, when applied, will help avoid errors.

3n = (degree of freedom)site = ~ f~ (3) ,y

3n = (degree of freedom)~aetor grouv = ~ a~C~, (4)

where a~ = ~ a~. (5)

(6) The irreducible representat ion of the crystal gives the number of lattice vibrat ion in each species of the factor group. The total irreducible representat ion of the crystal, F crystal, is the combined irreducible representat ion of each equivalent set of atoms, FEq" set"

The F~q. set is constructed in the following manner :

r E q . set • ~ a ~ - ~, ( 6 )

where at, as previously defined, is the number of lattice vibrations of the equivalent set of atoms in species ~- of the factor group. The total irreducible representat ion of the crystal r crystal c a n be constructed as follows :

r c r y s t a l __ ~ - - Eq. set 1 - [ - r E q . set 2 - ~ " " " ( 7 )

This irreducible representat ion of r Cry~tal contains the acoustical vibration. In the following examples, the acoustical vibrat ion will be removed from this representat ion by simple substracting out the irreducible representat ion of the acoustical vibrations, as follows :

Fcrystal __ p c r y s t a l paeous t ica l ( 8 ) vibrat ion - - x - - ~ •

N O W -perystal ±vibration is the irreducible representat ion of the lattice vibrations in the crystal.

This procedure needs only minor modification to include the intramolecular vibrat ion and libration for molecular crystals. Here the irreducible representation of a molecular crystal can be defined as r m o l e c u l a r crystal ~c rys t a l !

vibrat ion = ±v ib ra t i on -I- ~molecu la r vibrat ion

-'1- I~libration - - r ac°ustical- (9)

The molecular crystal NH4I, example 6, will demon- s t ra te the usefulness of Eq. (9).

Utilizing the above definition, the degrees of vibrational freedoms for each species of the site group D2a are tabulated in Table VI for the equivalent set of t i tanium atoms.

Table VI indicates the presence of the t i tanium lattice vibrat ion designated as degrees of freedom in species B2 and E. The next step is to correlate the B2 and E species of the site group D~d to the D4h factor group species. The correlation tables are given in


Table VI. Titanium atoms on site D~a. The degrees of vibrational freedom for each species (n = 2 atoms/equivalent set).

D~a species Translation U

Degree of vibrational

freedom .fv = n . U

A 1 0 0 A~ 0 0 B, 0 0 Bz T: 1 2 E T:. T,~ 2 4

Appendix IV for D2d to D4t,, and also presented in Refs. 9 and 10.

Ext rac t ing only a port ion of the correlation tables given in Appendix IV, we find the following relation- ship between the site and factor group species.

D2d Correlation , ' D 4h Site group > Facior .group

spee.ies C:' ~ C:" species

A~ ~ : Aj U

B~,,

A~ . .~ A ~

B~ ~ ~ A,,,

B2o

B~ ---.-.-...._ ~ B ~

A 2,,

E ~ ~ Eg

E~

Therefore, there is no need for reproducing a complete set of tables here. Appendix IV t rea ts only two such tables pr imari ly to show how they are derived and provide the proper basis for selection of the eorrec¢ correlation tables when two or more possibilities exist. Since only the site species B2 and E contain these translat ions which are the lattice vibrat ions in the crystal, the correlations relating these species to those species in this factor group are of immedia te interest. By integrat ing the site species which contains the translat ions into the factor group b y the correla- t ion tables, it is easy to identify these lattice vibrations in the factor group species. Table V I I a shows this correlation and identifies the species of the lattice v ibra t ion in the crystal.

The t i t an ium a tom ' s irreducible representat ion for the factor group is obtainable through Eq. (4)

r = ~ 2 a r ' L

where a r = ~ a~,, i.e., the number of vibrat ions in T

species ~'. Therefore, the species of the factor group which contains lattice v ibra t ion involving the t i t an ium

Table Vlla. Correlation for the lattice vibrations of the titanium atoms in TiO~ crystal between the site group D2~ and factor group D4h.

Correlation D4h Factor a~ D~ Site ) group

fv species, T C~' ~ CV' species, ~" C¢ ar=aB,+aE

2 (TD B2 - ~ ~ B,~ 1 1 1 0

A 2 . 1 1 1 0

E~ 2 1 0 1

a tom can be wri t ten as the following irreducible representat ion I~Ti,

F T i = I " B l o + i " A 2 ~ + I " E g + i " E ~ .

A check can be made a t this point for possible errors utilizing Eqs. (3) and (4),

degrees of v ibra t ional f reedom in site groups

= 3 n = 6 = ~ f ~ = 6 Y

degrees of v ibra t ional f reedom in factor groups

3 n = ~ a~C~.= 1 + 1 + 2 + 2 = 6

where n = 2.

O x y g e n A t o m s . Following this same procedure, outlined above, the irreducible representat ion, tony, can be obtained for the equivalent set of oxygen atoms. A s u m m a r y of the necessary information is t abu la ted in Table VI Ib .

Checks

Eq. ( 3 ) : ~ f ' = 3 n = 1 2 ]

Eq. (4) : ~ a~C~ = 3n = 12[ degrees of freedom.

~- J Table Vllb. Tabulations of terms and correlations necessary to calculate the lattice vibrations of the oxygen atom in Ti02 crystal.

Correlation D4h Factor a~ C~, Site ) group

.f~ H species ~ (C~.~d species ~ Cr ar =aA~+alh+aB2

4 1 ( T ~ ) A i ID Alg 1 1 = 1 -I- 0 + 0

~ . A2g 1 0 + 0 + 0 0 =

~ - - Bi, 1 1= 1 + 0 + 0

~ B2~ 1 o = o + O + 0

4 1 (T=)B| ~- - - ~ E o 2 2 = 0 + l + 1

~ \ " \ f / ~ A,, 1 0 = 0 + 0 + 0

\ / \ ~ A , , 1 1 = 1 + 0 + 0

~ ; [ [ 1 O= 0 + 0 + 0

4 1 (Tu)B2 1 1 = 1 + 0 + 0

~kE, 2 2 = 0 + 1 + 1


Table VIII. D~ factor group, translation, and acoustical modes.

Translation Acoustical D4~ Species species mode species

A ~ T~ 4

The number and species of oxygen lattice vibration can now be calculated, for Foxy = ~ a:. ~',

Pony= ~ a t . ~" = 1Ai~-4-OA2~-I-iBz~WOB2g+2Eo

+OAi,~-f- iA2u+OBz~+iB2~-4-2Eu (10)

collecting terms :

Fo~y= A lg + Bi~-4-2E o + A 2~-k B2~+ 2E~.

The total representation of the crystal, F crystal, c a n

be calculated using Eq. (7) where F crystal is the sum of the individual irreducible representation for each set of equivalent atoms, or

r e r y s t a l -~ 1 ~ Ti "~ I~oxy

FWiO, c r y s t a l = ( B zg + A 2,~-4- E g + E ~,) + ( A lg + B zo + 2E ~

+A~,+B2~+2Eu) =A~+2A2~+2Bz~+B2~+3Et,+3Eu.

Applying a check at this point on ~he vibrational degree of freedom, Eq. (4) gives 3n = ~ at. Cr where

n is the number of atoms in the Bravais cell, for TiO~ n=6 . Therefore 3 . n = 18= 1CA,o+2C~+2C~1~ +iCBz,+3C~,A-3C~.

The acoustical vibrations are included in the irreducible representation, F TiO~crystal, given above. Of the 3n degrees of vibrational freedom, three are acoustical modes. When one considers only those vibrations at the center of the Brillouin zone, i.e., k~0, the three acoustical vibrations have nearly zero frequency. Since vibrations with zero frequency are of no physical interest here, they can be substracted

from the irreducible representation as suggested in Eq. 8

l~erystal __ r e r y s t a l raeoustieal. vibration - -

The acoustical modes are readily identifiable in factor groups since they have the same character as the translation; Table VIII shows this identification.

Therefore, the irreducible representation of the acoustical vibration r ac°ustieal = A 2~A-Eu.

The results of this factor group analysis identifying the number of lattice vibrations in each species and the spectral activity are summarized in Table IX.

Table IX gives the following selection rules for first-order Raman and ir activity in the TiO2 crystal.

Raman spectrum: Six fundamental lattice vibrations are allowed A ~ ,+2B~+3E, . Infrared spectrum: Three fundamental lattice vibrations are allowed

A2~+2Eu.

One vibration will be inactive in both the ir and the Raman spectrum: B2~. Also, there will be no coinci- dences, i.e., the same vibration mode which is active and observable by both the Raman effect and in the ir spectrum.

This completes the original goal set forth to obtain (1) the number of lattice vibration in TiO2 (anatase) and (2) the spectral activity of these vibrations. Additional information may be obtained by studying the polarization direction of Raman scattering; however, this procedure will not be discussed here.

2. SrTiO3 Crystal

Now that the stepwise procedure has been applied to obtain the molecular vibrations and activity for Ti02 crystal; there are several short cuts which, when applied, reduce the calculation to but a few minntes. SrTi03 crystal will serve as an example of this simplified procedure.

Table IX. D4h factor group species, translations, acoustical modes, number of lattice vibrations, and ir and Raman activity of TiO2 crystal.

Raman D4h Factor Translation Acous~bical rTiO, crystal FF~ O* ir polarization Raman

group species species mode species coefficients coefficients ~ activity b tensor species activity ¢

A;a 1 1 A~g B~ 2 2 B~ Eg 3 3 Ai~ A ~ T, 1 2 1 Bzu B2~ 1 1 E,, T~, ~ 1 3 2

4

4

~zz ( ~ - ~ )

~zy CQyz~zx

pTiO2 _pcrystal TiO~ ~acoustical vib - -- 1

(Azo --~-2A2u --~2Bzg--~B2 u --t-3Eg +3E~) -- (A2~ +E~) =Aig+A2u-4-2B1g--~-B2~,-~-3Eg+2E.. These coefficients are the number of lattice vibrations present in the species.

b Only those species which contain the translations are infrared active, i.e., A2u and E . are the only species which have ir active vibration. ¢ Those species which contain the polarizability tensor can have Raman activity, i.e., Azg, Big, B2g, and Eo can have Raman active lattice vibrations. This infor-

mation is readily available from Refs. 9 and 10.


Pertinent Information.

Crystal: SrTiO~, Pm~m--0~ t, Z ' = 1 (see Table I). Equivalent atom site: Sr--O~(n= 1); Ti- -O~(n= 1);

oxy atoms--D4~(n=3). (see Table IV).

Irreducible representation of each equivalent set of atoms :

Strontium.

O. site s y m m e t r y

species Correlation Oa factor conta in ing group

fff l ~ t rans la t ion species Cr ar

3 3(T.,~,~) F,. ) F,,, 3 i

• ".Fs~= ~ a r . f = l . F ~ f

T i t a n i u m .

O~ site s y m m e t r y

species Corre la t ion Oh f a c t o r ) conta in ing group

.f* t * t rans la t ion species Cr ar

3 3(T~,. , . ) F~,, ) F , . 3 1

FTi= I ' F~ ,

O x y g e n .

.P

D ~ Site symmel ry

species Correlation Oh Fac to r a 7 ) contailf ing group

l ~ t rans la t ion species C r af=aA~+aE,,

3 [ ( T z ) A2t, ~ ~ F l , , ( T z , v , , ) :~ 2 = l l

6 2(T~.~,) E,, ~ " - - - - "~ F~,, 3 1 = 0 1

Foxy = ( 2 F ~ + F : , ) F ae°ust ieal = F 1 u

Summary.

FSrTiO: p S r ~_ F Ti _~_ F o x y _ Faeous t i ca l c rys t r i b

(F~)T(F~,,)+(2F,~+E:,,) ( 1 , )

The O, character tables, see Ref. 9, identify F l u a s

ir active and F2~ as neither ir nor Raman active. Therefore, SiTiOs has three ir active fundamental vibrations and no first-order Raman spectrum. Appen- dix I I I gives identical irreducible representation by a different, more laborious method.

3. Cu._,O Crystal

h f fo rma t ion : OP--P.~m, Z B = 2 (see Table I ) . Equivalent a tom site: Cu-D3h ( n = 4 ) ;

oxy-Td (n = 2) (see Table IV).

Irreducible representation of each equivalent set of atoms.

C o p p e r .

.f~

D3~ Site s y m m e t r y

species Correlation Oh F a c t o r av ) con ta in ing group

C t r ans l a t ion species C¢ ar=aA2+aB.

I(T~) A2,, ~ ~ A2,, 1 1 1 0

2 (7'~, ~) E,, "" ~ E . 9 1 0 l

Ft,, 3 '2 l 1

F2,, 3 1 0 t

Oxygen.

Fcu =A2~+E,~+2Fi,,+F2,,

7'd Site s y m m e t r y

species Correlation Oh F a c t o r conta in ing group

.U U t rans la t ion species Cr ar

6 3 (T~, ~,.) F2 ~ " F t . ( T ~ , ~ : t 3 t

F~ o :; 1

F o x y = F l u - ~ - F 2 e

r a c o u s t i e a l -~ Flu. ~Cu~O Irreducible representation of the crystal ~ib.

rcu~o vib = FCu+ Foxy-- I~aeoustical

= (A~,WE~W2F~,WF2u)+ (F~u+F2o)- ( ~ ' l u )

Spectral act ivi ty : l~aman--F2g ; i n f r a r ed - -F~ .

Therefore, there will be one F2g active fundamental lattice vibrat ion active in the Raman effect and two Fh, ir active lattice vibrations.

4. ~-AI20s Crystal

Informat ion : R~c-D~d6,:Z B = 2 (see Table I). Equiva- • lent a tom site : A1--C3 (n = 4) ; oxy--C2 (n = 6) (see Table IV).

Aluminum.

.f"

C3 Site s y m m e t r y D.~,t

species Correlation F a c t o r a~ conta in ing group

l ~ t rans la t ion species C¢ a ¢ = a A + a E

I(T:)

2(7'..,)

A 1 1 1 -{ -0 ' ~ A,~ =

A2u 1 1 = 1 + 0

E E,j '2 2 = ~l + 2

\ \ ~ , A~,, 1 t = I + 0

\ ~ A 2 ~ 1 1 = 1 + 0 \

~" E,, 2 "2 = (; + '2


Here one observes that aa and a~ have different values. Reviewing Eq. 3, f f = a ~ ~Cr, gives the values of

a~. The a~'s are evaluated in the following manner.

Site species A: fA=4=aA(CAi-I-CA~+CA~+Ca2~)

= a A ( 4 ) .'. a a = 1.

Site species E : f~ = 8 = a~ (C~, + C ~ )

= a'~ ( 2+2 ) .'. a~ =2.

Then FA~=A~+A~a+A~u+A~u+2Eo+2E~.

Oxygen.

C~ Site symmetry Dsa

species C o r r e l a t i o n F a c t o r a ~ ) containing group

f~ t ~ translation species C~ , a~=aATaB

6 i(T~) A ~ A~a 1 1 1 0

~ A ~ 1 2 0 2

~ ' ~ "U~ Ea 2 3 1 2

A~. 1 1 1 0

12 2 (T~. v) B A2~(T,) 1 2 0 2

"~" ""~E~,(T~,u) 2 3 1 2

Fo~y=Aao+2A~o+3Eo+A~,+2A~,+3E., and

F . . . . . ti¢al = A 2 . + E ~ ,

Summarizing, Eqs. (7) and (8) give

FAI~O, _ ( A l a + A 2 o + A i . + A 2 , + 2 E g T 2 E , ) r i b - -

(Alo+2A2g+3Eg+Alu+2A2~+3Eu - - (A2~+E~)

FAI~O,_ 2A lg¢R) + 3A 2alo ) + 2A I~ ¢R) + 2A 2u( o ) r i b - -

-~ 5Eo(ir) + 4 E u (jr)

Here one can indicate the ac t iv i ty of each species by a superscript as follows: ( R ) = R a m a n active; (ir) = infrared active; (0) = inactive. This information is available from the character tables given in Refs. 9 and 10.

5. ZrO2

Information: P2,/o--Csh 5, ZB=4 " (See Table I). Equivalent a tom site : Z r - -Ci (n = 4) ; Oxy--C1 (n = 8) (see Table I I I ) .

Zirconium.

f ~ t •

C~ Site symmetry

species Correlation C2h Factor containing group translation symmetry Cr a~

12 3(T~.,,.,) A. A. Ba

A .

B .

t 3

1 3

1 3

1 3

FZr=3Ab+3Bo+3A, ,+3B, ,

Oxygen

f ~ t ~

C~ Site symmetry

species containing translation

C o r r e l a t i o n C2h Factor group

symmetry Cr ar

24 3(T~,u,~) A • Aa 1 6

Bg 1 6

Au(%) 1 6

B~(T~,u) 1 6

Foxy=6Ag+6Bg+6Au-k6Bu

r . . . . . t i c a l = A u + 2B ~.

Summarizing,

r Z r 0 2 __ r i b - 9Aa(a)+9Bz(It)+8A~(i~) + 7Bu(i~).

6a. NH4I Crystal (phase III)

Durig and co-workers have made extensive investi- gations of the molecular crystal, NH4I. 6 This crystal not only possesses lattice vibrations but also l ibration and intramolecular vibrations of the NH4 group in the crystal. I t is worthwhile to repeat Durig's calcu- lations with some modifications to demonstra te the usefulness of the correlation method when applied to a molecular crystal.

A natural division in applying the correlation method to a molecular correlation can be made as follows. (1) Derive the lattice vibrat ion of the NIt4 + ions and iodine atoms. (2) Calculate the libration, i.e., rotation, of the NH4 + group in the crystal. (3) Use the correlation technique to predict the number of intramolecular vibrat ion of the NH4 + group. The lattice vibrations ~re sometimes referred to as external vibrations while the molecular vibrations within a crystal are called internal vibrations. By combining the irreducible representat ion obtained from parts 1-3 using Eq. (9), the total representat ion for NH4I is constructed.

6b. Lattice Vibrations of NHJ Ion and Iodine Atom

NH4 + ion. Informat ion: NH4 +, site D2a, Z B= 2 (see Table IV). N H 4 +

f~ t *

D2a Site symmetry Correlation 044

species ) Factor a~ containing C~'~ C:' group translation species Cr a~=aEWaB~

2 (Tx, Tu)

I(T,)

E - ~ ~ E~ 2 1 1 0

" ' " - ~ Eu 2 1 1 0

B2 ~ B2a 1 1 0 1

~ ' ~ ' ~ ' ~ A2u 1 1 0 1

F C • ' ~ C=' N H , :B2a(R)+A2u( i ' )+Eo (R)+Ea(ir)


The above correla t ion applies the tables der ived in Append ix V. Repea t ing a por t ion of this table, one realizes there are two possible correlat ions of D:e into D4h. These are given below in Tab le Xa .

Corre la t ion (1) C~' ~ C~' in Tab le X a was used in the above calculat ion, bu t w h a t would be the results if correla t ion (2) C : ' --~ C ~ " is used?

Repea t ing this calcula t ion bu t using the second corre la t ion given in Table X which maps D~d c, ' ~ c,'~ D4~.

N H 4 +

D~ Site symmetry Correlation D4h

species ~ Factor a~ containing C~' ~C2" group

f~ t~ translation species C i- a~ aE.+-as~

4 2(T,,~)

2 ~ (%)

E. ~ .~ Eg 2 1 1 0

" " ' " " ~ E, 2 1 1 0

B 2 ~ > B , ~ , 1 0 1

~ . 1 1 0 1

The results are

Corre la t ion (2) ~c~'-~ c~" J.. N H 4 + = Blo(R)--[-A 2u (jr)

+Eo(R)+E~( i~) ,

compar ing to the first ca lcula t ion

Corre la t ion (1) r ~ : ~ c~'=B2 (R)+A2 (i~)

+ E g ( ~ ) +E~,(i~).

B o t h irreducible representa t ions predic t two ir and two R a m a n act ive f u n d a m e n t a l v ibrat ions . I n this specific instance, an improper choice of the correla- t ion table does no t al ter the predic ted spectra l ac t iv i ty . This is n o t the general rule as the reader will find t h r o u g h experience. (For example, see Ref. 11).

Table Xa. Two possible correlations which map D2d into D~.

(1) (2) Factor group Cs ~ ~ C~ t C2' ~ C~"

D4~ D ~ Did

A2~ B~ B~ B2a Bs Bl Eo E E E , E E Big Bx B2

Table Xb. NH4 ÷ lattice vibrations. The result using two different correlations from Table Xa. The polarizability tensors are given for certain species of D4~ point groups.

Correlation Correlation Polarizability C2' --~ C~' C2" -~ C~' tensor~

• . . Bag (,~-o~uv) B2g • • • ~ x v

A2~ A2~ • - - Ea Ea auz,azx E . E , • • •

a See R e f . 9.

W h e n the results are c o m p a r e d using bo th correla-

t ions in Tab le X, a difference in the presence of the

polar izabi l i ty t ensor is no ted (see Tab le Xb) .

Summa r i z ing the differences found in Tab le X b ,

Big species con ta in the polar izabi l i ty tensor ( a ~ - a ~ )

while species B~g possesses a ~ . Of course, polar ized

R a m a n studies on this or iented crys ta l would de tec t this difference; however , the exper imenta l results are

ex t remely difficult to ob ta in for this phase of the crystal .

This mis take will no t occur if the proper selection

is made in the corre la t ion tab les ; however , the au thor s felt it useful to include an example of an improper

choice to acqua in t the reader wi th this problem.

Table XI. Brillouin zone for simple cubic space groups, O~ ~ (P~8,~) (see Ref. 5).

Coordinates of typical No. of point in special position

Wyckoff's equivalent x /a, y /a, z / a notation positions (a = side of cubic cell)

Notation of BSW" for corresponding point

in reciprocal space

Type of symnmtry at corresponding point in

reciprocal space

a 1 O, O, 0 F Oh b 1 9, 9, ½ R 0~ c 3 ½, ½, 0 M D4h d 3 9, 0, 0 X D4h e 6 u, 0, 0 ~ C4~ f 6 ½, 9, u T C4v g 8 u, u, u A Car h 12 ½, u, 0 Z C2, i 12 u, u, 0 E C~ j 12 ½, u, u S C~ k 24 O, u, v W C, l 24 ½, u, v G C, m 24 u, u, v F C,

B S W = B o u c k a e r t , S m o l u e h o w s k i ; a n d W i g n e r .


Lattice Vibrations of Iodine Atoms. Informat ion: I, site C4,., Z ~= 2 (Table IV). Iodine:

C~ Site symmetry D,~

species Correlation Factor a~ )

cont.aining group ,fv t ~ translation spec ies C i- a t = a A ~ + a E

'2 i(T~) A~ ~ l* A,~ 1 1 1 0

A.,.,,(T,) 1 1 1 0

4 2(T. . , ) E ~ *' Eo 2 1 0 I

E,,(T~-,v) 2 1 0 1

F1 = A 1, (") + A 2u (jr) + E , (R) + E . (i:)

l~ ...... tical = A 2,, + E,,

Summary of the lattice vibrations in the N H d crystal :

iaNttd ( B ~ o + A 2 ~ + E o + E . ) l a t t i c e r i b =

+ ( A t ~ + A 2 , + E g + E ~ , ) -- (A.,.u+E~)

I ~ N H 4 / l a t t i c e v ib = A l a ( R ) - ~ - A 2u ( i r ) +Bag (R) +2Eg (m q-Eu (it)

6c. Librations (rotations) on the NH4 + Ion in the Crystal

The rotations of the NH4 + ion about the x, y, or z axis have the same character as the rotations (R., Ry, and R,) contained in the character table :of the D2d site group. Therefore, the species for the librations parallel to the x, y, or z axis will be easily identifiable. The correlation method is now applied to relate the librations oa the site group to the specific species of the factor group. The following slight modifications are necessary in t reat ing librations.

(1) f , ' = ' a ~ E C~, (11)

where Ra~ is the degree of rotat ional freedom con- t r ibuted by ~ species of the site group. Also, Rat = ~ Ra x.

-/

(2) FHbratio.= ~ ~a~'~. (12)

Of course, one notes Eqs. (11) and (12) are identical to Eqs. (2) and (6), respectively, with only the superscript R added to indicate rotation.

NH~ Librations. Informat ion : NH~--s i te D:a, Z" = 2 (see Table IV).

D._,,t Sil.e D4h symmetry Correlation Factor av

species with c:' ~ c: '') group R ~ rotation species C~. a~=aA2+aE

2 ](R=

4 2(R.. v)

1 1 o

1 1 1 0

E - ~ ~ E~ 2 1 0 1

E. 2 1 0 1

(1) rrCg{-~c¢'=~ a ~ . ~ ' = A 2 g + B ~ u + E g + E . .

If the correlation tables C2' --~ C2' given in Table X were used here to map D2d into D4h, the following irreducible representat ion would be obtained:

F c • ' - - ' C e - A • + B 2 ~ , + E + E ~ r o t - - g g •

Comparing the above two irreducible representations for the librations, we note the following difference.

Fc~,-~c~' indicates ~ libration in species B2, while r o t

Fc¢'~c~' does not contain species B2, bu t has the r o t

additional species B~ . Therefore, the representat ion differs in the presence (or absence) of species B2, and B~ . Nei ther species B~, or B2~ is infrared or Raman active ; therefore, in this specific case the selec- t ion rules are unaffected by the choice of correlation table. The proper choice of the correct correlation would be impor tan t in predicting the spectral act ivi ty of overtones, combinations, and difference tone where the symmet ry of this l ibration must be correctly known.

The reader is referred to Durig's pape# for an excellent application of deuterat ion studies which distinguished between the lattice vibrat ion and the libration in tiffs crystal.

6d. Intramolecular Vibrations of NH4 ÷ Ion

The correlation methods can be used again to place the different intramolecular vibrations of the NH4 + group in the proper species of the factor group. First, the number of iutramolecular vibrations of the NH4 + ion can be obtained using Td molecular symmet ry of the individual ion. The irreducible representat ion is FF = A i + E + 2 F 2 . Then, these molecular vibrations are correlated to the D2a site species and the site species are integrated into the factor group in the following manner. Here, to avoid clutter, each species of Ta molecular symmet ry will be dealt with separately.

A new column, Vvlb, is introduced which is the degree of vibrat ion freedom of the single ion NH4 +. For these molecular vibrations, the following summat ion is used

Z • vvib, = 3 n - - 6 = 9 = l ( A 1 ) + 2 ( E ) - + 6 ( F 2 ) = 9

also

f 'Y ~ Z ° P v i b ,

where Z B = number of NH~ + molecule in the Bravais cell; therefore, f f becomes the degree of vibrat ional freedom in the Bravais cell.

In format ion Needed. NH4 + ion Td molecular symmetry, D2~--site symmetry, and D4h factor group symmetry . Z B= 2 (see Tables I and IV).


C o r r e l a t i o n C o r r e l a t i o n )

) C g ~ C : " f T ~ Z " IIv ib

Molecular symmetry

~vib Td

Site symmetry

D~d C~ a i-

D4h Factor group

species Cr a i-

2 1 A~

4 2

i2 6

a l 1 22 ~ A,~ B2~

B2.

B, 1 2 ~ B,g A,,,

F2 ~ . . . . ~ E 2 4 ~ E~ Eu(T~.v )

B.., 1 4 ~ ~ B~,,

A2~(Tz)

1 I

1 1

1 l

1 l

1 l 1 l 2 2 2 2

1 2

1 2

The irreducible representat ion and spectral ac t iv i ty can be summar ized in the following manner .

Trans- Acous- Rota- Crystal lations tical tion Intra- Spectral

symmetry (lattice vibra- (libra- molecular activity D~ vibration) tion tion) vibration D4h

Azo 1 2 R A~u 1 • • ' B,g 1 R Bag l 2 R Eg 2 1 2 R Azu * • • A~ 1 1 2 ir B,~ 1 0 • • • B~ 2 • • • E~ 1 1 1 2 ir

Final check: Total vibrations (3N)=9+3+6+18=36, where N=12.

II. GENERAL APPLICATION OF CORRELATION METHOD

The authors and friends have successfully applied the basic procedure presented herein to m a n y crystals. Only ionic type crystals have been t rea ted in this paper ; however, no problem was encountered in

predict ing the selection rules for graphite, diamond, and polyethylene. The beau ty of this method lies in the basic applicat ion of simple principles in group theory. The only errors which can occur will be our own. The authors feel tha t this method adds still another degree of en joyment in doing molecular

spectroscopy !

III. SELECTION RULES FOR k ~ O

Previously, we have discussed only those selection rules for the center of the Brillouin zone, i.e., k = O .

At this boundary, k = 0, the acoustical vibrat ions were disregarded because of their near-zero frequency. The acoustical vibrat ions are not near zero a t other

points in the Brillouin zone and must be considered in the irreducible representat ion for these points in reciprocal space. Table X I gives a description of the Brillouin zone nomencla ture for the cubic space group 0h 1 and the nota t ion for the corresponding points in reciprocal space is shown in Fig. 1. To obtain the irreducible representat ion for some point in reciprocal space, we need only to correlate the lattice vibrat ions occurring in the site group species to the type of sym- me t ry at the corresponding point in reciprocal space; a typical example of these symmetr ies is given in Table X I for the cubic space groups 0h 1. Now the acoustical vibrat ions are not subs t rac ted f rom the irreducible representat ion, but mus t be included because of their nonzero frequencies.

The selection rules for ir and R a m a n act iv i ty of fundamenta l vibrations, acoustical vibrations, overtones, different tones, and combinations are deter- mined f rom the point s y m m e t r y group in reciprocal space. Care must be exercised in this application.

FIG. 1. The Brillouin zone for 0hi(Presto). Noiations from Bouckaert, Smoluehowski, and Wigner. Side of the cube equals 1/a (Ref. 5).


ACKNOWLEDGMENTS

First, we wish to thank Professor Jim Durig for his patient instructions in the basic details of the correlation method. Many useful discussions with Dr. G. L. Carlson, Dr. R. J. Van Dyke, Professor William White, and Professor Foil A. Miller have encouraged us to write this review paper, adding modifications at points which appeared to need clarity.

One of the authors (W. G. F~teley) wishes to acknowledge the partial support of Air Force Contract No. F33615-70-C-1382 and Mellon Institute.

APPENDIX I

Site Symmetry Table for Bravais Space Cell

The following table was compiled from the site symmetry tables for crystallographic groups found in the International Tables for X-Ray Crystallography, Vol. I. Please note that the table is modified and should only be used for the Brav~is space cell and the Halford site symmetry correlation method suggested in this article. The site symmetries are arranged in alphabetical order reading from left to right. See Appendix II for an explanation of this ordering.

Space group Site symmetr ies

1 P1 C, ~ C1(1) 2 P I C~' 8C~(1) ; C,(2) 3 P2 C2' 4C2(1) ; C,(2) 4 P2, C~ 2 C, (2) 5 B~ or C~ C~ ~ 2C~ (1); C, (2) 6 Pro" C. ~ 2C~ (1) ; C, (2) 7 Pb or Pc C~ ~ C,(2) 8 B m or Cm C~ ~ C,(1) ; C,(2) 9 Bb or Cc C, 4 C,(2)

10 P 2 / m C ~ ' 8C2~ (1) ; 4C~(2) ; 2C.(2) ; C,(4) 11 P 2 , / m C~ ~ 4C~(2) ; C~(2) ; C,(4) 1 2 B 2 / m o r C 2 / m C:~ ~ 4 C ~ ( 1 ) ; 2 C i ( 2 ) ; 2 C ~ ( 2 ) ; C,(2);

C, (4) • 13 P 2 / b or P 2 / c C2h 4 4C~ (2); 2C~(2) ; C, (4) ., 14 P 2 , / b or P 2 , / c C2~ ~ 4Ci(2) ; C,(4)

15 B 2 / b or C2/c C ~ ~ 4C~ (2) ; C~ (2) ; C, (4) 16 P222 D~' 8D~(1) ; 12C2(2) ; C~(4) 17 P222, D2 ~ 4C~ (2) ; C1 (4) 18 P2,2~2 D2 ~ 2C~(2) ; C~(4) 19 P2~2~2~ D~ ~ C1 (4) 20 C222, D2 ~ 2C2 (2) ; C, (4) 21 C222 D26 4D2(1) ; 7C~(2) ; C~(4) 22 F222 D~ ~ 4D~(1) ; 6C~(2) ; C~(4) 23 I222 D2 s 4D~(1) ; 6C~(2) ; C~(4) 24 I2~2,21 D2 ~ 3C~ (2) ; C1 (4) 25 Prom2 C~. ~ 4C~(1) ; 4C.(2) ; C~(4) 26 Pmc2~ C~. ~ 2C.(2) ; C~(4) 27 Pcc2 C2~ 3 4C~(2) ; C1(4) 28 Pma2 C~ 4 2C~ (2) ; C~ (2) ; C, (4) 29 Pea% C ~ ~ C, (4) 30 Pnc2 C~v ~ 2C~(2) ; C,(4) 31 Pmn2~ C2J C, (2) ; C, (4) 32 Pba2 C~ s 2C~ (2) ; C, (4) 33 Pna2~ C ~ ~ C,(4) 34 P n n 2 C ~ ~ 2C~(2) ; C,(4) 35 Cram2 C2. u 2C~(1) ; C~(2) ; 2C.(2) ; C,(4) 36 Cmc2~ C~o ~ C,(2) ; C1(4) 37 Ccc2 C ~ 13 3C~ (2) ; C, (4) 38 A t o m 2 C2v '4 2C~(1) ; 3C.(2) ; C~(4) 39 A b m 2 C~. '~ 2C~(2) ; C.(2) ; C,(4) 40 Area2 C~. '~ C~(2) ; C.(2) ; C~(4) 41 Aba2 C~,, '~ C2(2) ; C~(4) 42 From2 Car TM C~,(1) ; C~(2) ; 2C.(2) ; C1(4) 43 Fdd2 C~. ~ C~(2); C~(4) 44 I m m 2 C2,, 2o 2C~.(1) ; 2C,(2) ; C~(4) 45 Iba2 C~,, 2' 2C~(2) ; C,(4) 46 I m a 2 C ~ 2~ C~(2) ; C.(2) ; C~(4) 47 P m m m D~a ~ 8D~h(1); 12C~,(2) ; 6C~(4) ; C1(8) 48 P n n u D,a ~ 4D~ (2) ; 2C~ (4) ; 6C~ (4) ; C, (8) 49 Pccm D2a ~ 4Cab (2) ; 4D2 (2) ; 8C, (4) ; C. (4) ;

C, (8) 50 Pban D ~ ~ 4D~(2) ; 2C~(4) ; 6C~(4) ; C,(8) 51 P m m a D~n ~ 4C~a(2); 2C2~(2); 2C2(4); 3C.(4);

C,(8) 52 P n n a D ~ ~ 2C~(4) ; 2C~(4) ; C,(8) 53 Pinna D~a ~ 4C~(2) ; 3C~ (4) ; C, (4) ; C1 (8) 54 Pcca D:~ ~ 2C~(4) ; 3C~(4); C~(8) 55 Pbam D2~ ~ 4 C ~ (2) ; 2C~(4) ; 2C.(4) ; C~(8) 56 Pccn D2~ '° 2C~(4) ; 2C~(4) ; C~(8) 57 Pbcm D ~ u 2C~(4), C~(4) ; C~ (4) ; C, (8)

Space group Site symmetr ies

58 P n n m D2h 1~ 4C2h(2) ; 2C2(4) ; C~(4) ; CI(8) 59 P m m n D2h TM 2C2. (2) ; 2Ci(4) ; 2C~ (4) ; C1 (8) 60Pbcn D2h '4 2 C i ( 4 ) ; C 2 ( 4 ) ; C , ( 8 ) 61 Pbca D2h 15 2C~(4) i C1(8) 62 P u m a D~h 16 2Ci(4) ; C. (4) ; C~(8) 63 Cmcm D~h 1~ 2C2h(2) ; C2v(2) ; Ci(4) ; C2(4) ;

2C~(4) ; C~(8) 64 Cmca D~h TM 2C2h (2) ; Ci(4) ; 2C2(4) ; C~(4) ; C,(8) 65 C m m m D2h 1~ 4D2h (1) ; 2C~h(2) ; 6C2.(2) ; C2(4) ;

4C~ (4) ; C, (8) 66 Cccm D2h 2o 2D~(2) ; 4C~(2) ; 5C~(4) ; C.(4) ;

C, (8) 67 C m m a D~h 21 2D2(2) ; 4C~ (2) ; C2~ (2) ; 5C2(4) ;

2C~(4) ; C,(8) 68 Ccca D2h ~ 2D~(2) ; 2Ci(4) ; 4C~(4) ; C1(8) 69 F m m m D2n 23 2D2h (1) ; 3C2h (2) ; D2(2) ; 3C~(2) ;

3C2(4) ; 3C~(4) ; C~(8) 70 Fddd D2h 2~ 2D2(2); 2C¢(4); 3C:(4) ; C,(8) 7 1 1 m m m D2h 25 4D~h(1) ;6C2~(2) ;Ci (4 ) ;3C. (4 ) ;

C,(8) 72 I b a m D2a 2~ 2D~(2) ; 2C~(2) ; Ci(4) ; 4C~(4) ;

c~(4) ; c~(8) 73 Ibca D ~ ~ 2Ci(4) ; 3C~(4) ; C,(8) 74 I m m a D ~ 2s 4C~ (2) ; C2~(2) ; 2C2(4) ; 2C,(4) ;

C, (8) 75 P4 C, ' 2C4 (1) ; C~ (2) ; C, (4) 76 P4, C, ~ C,(4) 77 P4~ C4 ~ 3C~(2) ; C,(4) 78 P4a C44 C, (4) 79 14 C~ ~ C4(1) ; C~(2) ; C1(4) 80 I4 , Ca ~ C2(2) ; C,(4) 8l P4 $4 ~ 4S4(1) ; 3C2(2) ; C,(4) 82 I4 $42 4S4(1) ; 2C~(2) ; C,(4) 83 P 4 / m C4~' 4C4~(1) ; 2C~(2) ; 2C4(2) ; C2(4) ;

2C~ (4) ; C, (8) 84 P4~/m C~ ~ 4C~(2) i 2S~(2) ; 3C2(4) ; C~(4) ;

C1 (8) 85 P 4 / n Cab 3 2S~(2) ; C4(2) ; 2C~(4) ; C~(4) ; C,(8) 86 P4~/n C~a ~ 2S~ (2) ; 2C~ (4) ; 2C~ (4) ; C, (8) 87 I 4 / m C ~ ~ 2 C ~ (1) ; C2a (2) ; S~ (2) ; C~ (2) ;

e l (4) ; C2(4) ; C.(4) ; C,(8) 88 I4~/a C~h ~ 2S4(2) ; 2Ci(4) ; C~(4) ; C~(8) 89 P422 D4' 4D4(1) ; 2D~(2) ; 2C4(2) ; 7C~(4) ;

C,(8) 90 P42~2 D~ 2 2D:(2) ; Ca(2) ; 3C2(4) ; C~(8) 91 P4,22 D~ ~ 3C~(4) ; C~(8) 92 P4~2~2 Da ~ C2(4) ; C~(8) 93 P4222 D4 ~ 6D~ (2) ; 9C~ (4) ; C, (8) 94 P4~2,2 D4 ~ 2D~(2) ; 4C~(4) ; C1 (8) 95 P4a22 D J 3C~(4) ; C,(8) 96 P4~2,2 D4 s C:(4) ; C,(8) 97 •422 D49 2D4(1) ; 2D~(2) ; C4(2) ; 5C~(4) ;

C,(8) 98 I4~22 Da ~0 2D~(2) ; 4C~(4) ; C,(8) 99 P 4 m m C4~,1 2C4,(1) ; C2~(2) ; 3C~(4) ; C1(8)

100 P4bm CaJ C~(2) ; C~.(2) ; C.(4) ; C~(8) 101 P42cm C4J 2C~(2) ; C~(4) ; C.(4) ; C~(8) 102 P4~nm C~ ~ C~(2) ; C~(4) ; C.(4) ; C~(8) 103 P4cc C ~ 5 2C4(2) ; C~(4) ; C,(8) 104 P4nc C4~ ~ C4(2) ; C2(4) ; C~(8)


Space group

105 P4~mc 106 P4~bc 107 I4mm 108 I4cm 109 I4m~d 110 I4.cd 111 P42m

112 PY~2c 113 P.~2~m 114 P42~c 115 pTtm2

116 pTic2 117 PY~b2 1.18 PY~n2 119 I4m2

120 I4c2 121 I42m

122 142d 123 P4/mmm

124 P4/mcc

125 P4/nbm

126 P4/nnc

127 P4/mbm

128 P4/mnc

]29 P4/nmm

130 P4/ncc

131 P4~/mmc

132 P4~/mcm

133 P42/nbc 134 P4~/nnm

135 P4~/mbc

136 P4~/mnm

i37 P42/nmc

138 P4~/ncm

139 I 4 / m m m

140 I4/mcm

141 I4, /amd

142 I4~/acd 143 P3 144 P3~ 145 P3~ 146 R3 147 P 3 148 R3 149 P312 150 P321 151 P3112 152 P3~21 153 P3212 154 P3~21 155 R32 156 P 3 m l 157 P31m 158 P3c l

Site symmet r i e s

C4j 3C~(2) ; 2C~(4) ; C~(8) C4, s 2C~(4) ; C~(8) C4~ ~ C4~(1) ; C~(2) ; 2C~(4) ; C~(8) C4~ 1° C4(2) ; C~(2) ; C~(4) ; C~(8) C4~ n C2~ (2) ; C~ (4) ; C~ (8) C4~ '~ C~(4); C,(8) D~a ~ 4 D ~ (1) ; 2D~(2) ; 2C~(2) ; 5C~(4) ;

C~ (4) ; C~ (8) D~.? 4D2(2) ; 2S4(2) ; 7C~(4) ; C.(8) D ~ a 2S4(2) ; C~,(2) ; C2(4) ; C.(4) ; C1(8) D2d 4 2S4(2) ;2C~(4) ; C~(8) D~a ~ 4D~a (1) ; 3C~,(2) ; 2C~(4) ; 2C~(4) ;

C~ (8) D~d ~ 2D~(2) ; 2S4(2) ; 5C~(4) ; C,(8) D~d ~ 2S4(2) ; 2D~(2) ; 4C(4) ; Ct(8) D~d s 2S4(2) ; 2D2(2) ; 4C~(4) ; C1(8) D~d ~ 4D2~(1) ; 2C~(2) ; 2C2(4) ; C~(4) ;

C~ (8) D2d ~o D~(2) ; 2S4(2) ; D~(2) ; 4C~(4) ; C1(8 ) D~a n 2D~(1) ; D~(2) ; $4(2) ; Car(2) ;

3Cz(4) ; C~(4) ; C,(8) D~a '~ 2S4(2) ; 2C~(4) ; C~(8) D ~ ~ 4D4~ (1) ; 2D~a (2) ; 2C4~(2) ; 7C2.(4) ;

5Cs (8) ; C1 (16) D4a ~ D4 (2) ; C4~ (2) ; D4 (2) ; C4a (2) ;

C~a(4) ; D2(4) ; 2C4(4) : 4C2(8) ; C~(8) ; C~(16)

D4~ ~ 2D4(2) ; 2D~a(2) ; 2C~a (4) ; C4(4) ; C~.(4) ; 4C~(8) ; C~(8) ; C~(16)

Dab 4 2D4 (2) ; D~ (4) ; $4 (4) ; C4 (4) ; C~ (8) ; 4Ce(8) ; C~(8)

D~a ~ 2C4~ (2) ; 2 D ~ (2) ; C~(4) ; 3C~(4) ; 3C~(8) ; C~(16)

D ~ ~ 2C4~ (2) ; C~ (4) ; D2 (4) ; C4 (4) ; 2C2(8) ; Cs(8) ; C1(16)

D 4 J 2D~a(2) ; C4~(2) ; 2C2a (4) ; C~(4) ; 2c~(8) ; 2c~(8) ; c~(16)

D ~ s D~(4) ; $4(4) ; C4(4) ; Ci(8) ; 2C~(8) ; C, (16)

D ~ 9 4 D ~ ( 2 ) ; 2 D ~ ( 2 ) ; 7 C ~ ( 4 ) ; C ~ ( 8 ) ; 3C~(8) ; C,(16)

D4h I° D~h (2) ; D~d(2) ; D~h (2) ; D~a(2) ; D2(4) ; C~h (4) ; 4C~,(4) ; 3C~(8) ; 2C~(8) ; C,(16)

D~fl ' 3 D : ( 4 ) ; $ 4 ( 4 ) ; C ~ ( 8 ) ; 5 C ~ ( 8 ) ; C ~ ( 1 6 ) D4~ 1~ 2 D e ~ ( 2 ) ; 2 D ~ ( 4 ) ; 2 C ~ ( 4 ) ; C 2 " ( 4 ) ;

5C~(8) ; C~(8) ; C,(16) D4~ ~ C ~ ( 4 ) ; S ~ ( 4 ) ; C ~ ( 4 ) ; D ~ ( 4 ) ;

3C2(8) ; C.(8) ; C1(16) D4~ TM 2D2~(2); C~h(4); $4(4) ; 3C~(4 ) ;

C~(8) ; 2C~(8) ; C~(16) D~a ~ 2D2~(2 ) ;2C~ , ( 4 ) ;C~(8 ) ;C~(8 ) ;

C~(8) ; C~(16) D ~ ~6 D ~ ( 4 ) ; $ 4 ( 4 ) ; 2 C 2 ~ ( 4 ) ; C 2 ~ ( 4 ) ;

3c~(8) ; c.~(8) ; c,(16) D4h ~7 2D~h (1) ; D ~ (2) ; D ~ ( 2 ) ; C4v(2) ;

C~ (4) ; 4C~, (4) ; C~ (8) ; 3C~ (8) ; C~ (16)

D ~ ~s D4(2); D ~ ( 2 ) ; C4~(2); D ~ ( 2 ) ; C~ (4) ; C4 (4) ; 2C~, (4) ; 2C~ (8) ; 2C~(8) ; C,(16)

D ~ 19 2 D ~ d ( 2 ) ; 2 C ~ ( 4 ) ; C ~ ( 4 ) ; 2 C ~ ( 8 ) ; Cd8); C,(16)

D~a ~° $ 4 ( 4 ) ; D ~ ( 4 ) ; C ~ ( 8 ) ; 3 C ~ ( 8 ) ; C 1 ( 1 6 ) C3 ~ 3C3(1) ; C4(3) C~ 2 C~ (3) C. ~ C~ (3) c~4 c~ (1); c~(3) C3~ ~ 2C,~(1) ; 2C~(2) ; 2C~(3) ; C~(6) C37 2C,~(1) ; C,(2) ; 2C~(3) ; C1(6) Da ~ 6D, (1) ; 3C~ (2) ; 2C~(3) ; C~(6) D3 ~ 2D3 (1) ; 2C~ (2) ; 2C~(3) ; C1(6) D~ ~ 2C~(3) ; C~(6) D34 2C: (3) ; C~ (6) Da 5 2C~(3) ; C~(6) D, 6 2C~(3) ; C~(6) D3 ~ 2Da(1) ; C3(2) ; 2C~(3) ; C,(6) C3~ ~ 3C~(1) ; C.(3) ; C~(6) C.~ ~ C~(1) ; Cs(2) ; C,(3) ; C1(6) C~" 3C3(2) ; C~(6)

Space group

159 P31c 160 R3m 161 R3c 162 P31m

163 P31c

164 P 3 m l

165 P3c l

166 R3m

167 R3c

168 P6 169 P6~ 170 P6~ 171 P6~ 172 P64 173 P6~ 174 P 6 175 P6/m

176 P6a/m

177 P622

178 P6~22 179 P6~22 180 P6222 181 P6422 182 P6~22 183 P6mm

184 P6cc 185 P63cm 186 P6amc 187 P6m2

188 P6c2

189 P62m

190 P62c

191 P6/mmm

192 P6/mcc

193 P6~/mcm

194 P6~/mmc

195 P23

196 F23 197 I23 198 P2~3 199 I2 ,3 200 Pm3

201 Pn3

202 Fro3

203 Fd3

204 Im3

205 Pa3 206 Ia3 2O7 P432

208 P4~32

Site symmet r i e s

c~v 4 2c~(2); c , ( 6 ) c ~ ~ c~v(1); c , (3) ; c,(6) c ~ ~ c , (2) ; c~(6) Dad' 2D~a(1) ; 2D3(2) ; C3~(2); 2C2h (3) ;

C3(4) ; 2C2(6) ; C~(6) ; C1(12) D3a 2 D3 (2) ; C31 (2) ; 2D3 (2) ; 2C~ (4) ;

C~(6);C~(6);C1(12) D3a 3 2D3d(1) ; 2C3~(2) ; 2C2h (3) ; 2C2(6) ;

c~(6) ; c1(12) D3~ 4 D~ (2) ; C3~(2) ; 2C3 (4) ; Ci(6) ;

C2(6) ; C~(12) D3d 5 2D3d (1) ; C~ (2) ; 2C2h (3) ; 2C~ (6) ;

c , (6 ) ; c , (12) D3e 6 D~ (2) ; C3~(2) ; C3 (4) ; C~(6) ; C2(6) ;

C, (12) C61 C6(1) ; C3(2) ; C2(3) ; C,(6) C6 ~ C~(6) C63 C1 (6) C6 4 2C~ (3) ; C~ (6) C65 2C2(3) ; C1(6) C6 6 2C~ (2) ; C, (6) Cab ~ 6C3h(1) ; 3C3(2); 2C.(3) ; C1(6) C6h ~ 2C6h(1) ; 2Cob (2) ; C6(2) ; 2C~h (3) ;

C3(4) ; C2(6) ; 2C~(6) ; C~(12) C6h ~ C3a (2) ; C3~ (2) ; 2C3h (2) ; 2C3 (4) ;

C~(6) ; C~(6) ; C~(12) D61 2D6(1) ; 2D3 (2) ; C6(2) ; 2D~(3) ;

c , (4 ) ; 5c2(6) ; C~(12) D62 2C~(6) ; C1(12) D6 a 2C2(6) ; C1(12) D64 4D~(3) ; 6C2(6) ; C~(12) D6 ~ 4D~(3) ; 6C~(6) ; C,(12) D66 4Da (2) ; 2C:. (4) ; 2C:(6) ; C1 (12) C6~ ~ Car(l) ; C3v(2) ; C ~ ( 3 ) ; 2C~(6) ;

C~(12) C6. ~ C6(2) ; C~ (4) ; C~(6) ; C~ (12) C ~ a C.~(2) ; C~(4) ; C~(6) ; C,(12) C~ 4 2C~(2) ; C~(6) ; C~(12) D~h I 6D~(1) ; 3Ca~(2); 2C2,(3) ; 3C,(6) ;

C1(12) D~h ~ D~ (2) ; C:~ (2) ; D~ (2) ; C~ (2) ; Da (2) ;

C~(2) ; 3C~(4) ; C~(6) ; C~(6) ; C,(12) D ~ a 2 D ~ ( 1 ) ; 2 C ~ ( 2 ) ; Ca~(2); 2C~.(3);

Ca(4) ; 3C~(6) ; C~(12) D ~ 4 D~ (2) ; 3C~ (2) ; 2C~ (4) ; C2 (6) ;

C~(6) ; C~(12) D~a I 2D6~ (1) ; 2D~a (2) ; C~(2) ; 2D~ (3) ;

C~(4) ; 5C~v(6) ; .4C~(12) ; C~(24) D6~ ~ D6(2) ; C6h(2) ; D~(4) ; C~ (4) ; C~(4) ;

D~ (6); C~ (6); C~ (8); 3C~ (12); C~(12) ; C,(24)

D ~ 3 D ~ (2) ; D~d(2) ; C.~(4) ; D3 (4) ; C6(4) ; C~ (6); C~v(6) ; CC,(8) ; C~(12) ; 2C~(12) ; C, (24)

D6~ ~ D ~ (2) ; 3D~ (2) ; 2C~(4) ; C2~ (6) ; C~(6) ; C~(12) ; 2C~(12) ; C~(24)

T ~ 2T(1) ; 2D~(3) ; Ca(4) ; 4C~(6) ; C~ (12)

T ~ 4T(1) ; C~(4) ; 2C~(6) ; C~(12) T 3 T (1 ) ; D~(3); C~(4); 2C~(6); C1(12) .7 TM C~ (4) ; C~(12) T ~ C~(4) ; C~(6) ; C~(12) T~ ~ 2T~(1) ; 2 D ~ (3); 4C~,(6) ; Ca(8) ;

2C.(12) ; C. (24) T~ ~ T(2) ; 2C.~(4) ; D:(6) ; C~(8) ;

2C~(12) ; Ct (24) T~ ~ 2T~(1) ; T (2 ) ; C~ (6) ; C~(6) ; C~(8) ;

C~(12) ; C~(12) ; C1(24) Ta 4 2 T ( 2 ) ; 2C~(4) ; C~(8) ; C~(12) ;

C~(24) Ta b T~(1) ; D ~ (3) ; C~i(4) ; 2C2,(6) ;

C~(8) ; C.(12) ; C,(24) Th ~ 2C~i(4) ; C~(8) ; C1(24) T~ ~ 2Ca~(4) ; Ca(8) ; C2(12) ; C~(24) 0 ~ 20(1) ; 2D4(3) ; 2C4(6) ; C~(8) ;

3C~(12) ; C~(24) 0 ~ T (2) ; 2Da (4) ; 3D~ (6) ; C~ (8) ;

5C~(12) ; C~(24)


Space group Site symmetries

209 F432

210 F4~32

211 •432

212 P4~32 213 P4~32 214 14,32

215 PY~3m

216 F43m

217 143m

218 p743n

219 F3~3c

220 I7~3d 221 Pm3m

0 ~ 20(1); T(2); D~(6) ; C~(6) ; C~(8) ; 3C~(12) ; C~(24)

0 ~ 2T(2) ; 2D~ (4) ; C~ (8) ; 2C~(12) ; CC~ (24)

0 ~ O(1);D,(3);Da(4);D~(6);C~(6); Ca(8) ; 3C~(12) ; C~ (24)

0 ~ 2Da(4) ; C~(8) ; C~(12) ; C~(24) 0 ~ 2D~ (4) ; C~(8) ; C~(12) ; C~(24) 0 s 2D~(4) ; 2D~(6) ; Ca(8) ; 3C~(12) ;

Ct (24) T~ ~ 2To(l) ; 2D~(3) ; C~(4) ; 2C~(6) ;

C~(12) ; C,(12) ; Ct (24) T~ ~ 4Ta(1) ; C**(4) ; 2C~(6) ; C,(12) ;

C~(24) T~ ~ T~ (1) ; D:~ (3) ; C~. (4) ; S~ (6) ;

C~(6) ; C~(12);C,(12);C1(24) Td 4 T (2) ;D, (6) ; 2S~ (6) ; C~ (8) ; 3C~(12) ;

C~(24) Ta ~ 2T(2) ; 2S, (6) ; C~ (8) ; 2C~(12) ;

C~(24) T~ ~ 2S~(6) ; C~(8) ; C~(12) ; C~(24) O~ ~ 20a (1) ; 2D~a (3) ; 2C**(6) ; Ca~ (8) ;

3C~,(12) ; 3C~(24) ; C~(48)

Space group Site symmetries

222 Pn3n

223 Pm3n

224 Pn3m

225 Fm3m

226 Fm3c

227 Fd3m

228 Fd3c

229 Im3m

230 Ia3d

Oa ~ 0(2) ; D4(6) ; Cai(8) ; $4(12) ; C~(12) ; Ca(16) ; 2C~(24) ; C~(48)

0a a T~ (2) ; D~ (6) ; 2D~ (6) ; D.~ (8) ; 3Cz,(12) ; Ca(16) ; C~(24) ; C~(24) : C~(48)

0~ ~ T~(2) ; 2D~(4) ; D~ (6) ; C~,(8) ; D~(12) ; C~(12) ; 3C~(24) ; C~ (24) ; C~ (48)

O~ ~ 20~(1) ; Td(2); D~(6); Car(6) ;, C~,(8) ; 3C~,(12) ; 2C~(24) ; C~(24) ; C~ (48)

Oa ~ 0(2); T~(2); D~,~(6); C~(6) ; C~,(12) ; C~(12) ; C~(16) ; C~(24) ; C~ (24) ; C~ (48)

0~ 2T~(2) ; 2D~(4) ; C~,(8) ; C~,(12) ; C,(24) ; C~(24) ; C~(48)

O~ s T (4) ; D~ (8) ; C~ (8) ; S~ (l 2) ; Ca (16) ; 2C~(24) ; C~ (48)

0~ 9 0~ (1) ; D~ (3) ; D~ (4) ; D~,~(6) ; C4~(6) ; Ca,(8) ; 2C~v(12) ; C~(24) ; 2C~ (24) ; Ct (48)

O~ TM C~(8) ; Da(8) ; D~(12) ; $4(12) ; C~(16) ; 2C~(24) ; C,(48)

N o t e the following equ iva len t nomenc la tu re s : C~ ~ $2;

C~-=Clh; D 2 ~ V ; D 2 h ~ V h ; D 2 d ~ V d ; C~i------S6.

A P P E N D I X II

Site Symmetry

The first example t h a t had an ambiguous choice of

~he site s y m m e t r y was the equ iva len t set of oxygen

a toms in Ti02. Recal l ing f rom the x - ray in fo rma t ion

t h a t four equ iva len t oxygen a toms are present in the

Brava is cell, this equ iva len t set could be placed ei ther on C2~(4) or C2~(4) sites. Ac tua l ly the Wyc kof f

tables s on the publ ished crys ta l lographic d a t a indicate

the site posi t ion of each equiva len t set of a toms in the

following nota t ion .

Wyckoff's Tables for TiO2.

Atom Site Position

Ti (a) 0, 0, 0; 0, b, ½ Oxygen (e) 0,0, u ; 0 , 0 , ~ ; 0 , ½ , u+¼; 0, ½, ¼ -u

One could consul t the c rys ta l lographic tables 7 and

ident i fy the site f rom the x, y, z coordinates ; however , a m u c h s impler p rocedure can be followed. Append ix I

presents the site s y m m e t r y in the a lphabet ica l order in t h e following manner . Using TiO2 as an example,

Append ix I gives for D4h~9: 2D~(2 ) , 2C2h(4), C2~(4),

2C2(8), C~(16). N o t i n g t h a t D2d, C2h, and C: appea r twice, one can write f rom the da t a in Append ix I the

following a lphabet ica l ordering.

Alphabetical ordering

Site in Alphabetical of site Atom Appendix I order position (site) ~

2D2d (2) D2~ (2) a D~d(2) b

2C~h (4) C~h (4) c C2h (4) d

C2,(4) C~,(4) c 2C2(8) C2(8) f

C~(8) g C~(8) C,(8) h C,(I6) C,(16) i

Ti (a)

()xy (e)

Informat ion from Wyckoff's table, and the references.

T h e a lphabet ica l le t ter in the parenthes is fol lowing Ti in Wyckof f ' s table indicates the site of the a tom, i.e., (a) indicates the t i t a n i u m is on site D2d. F o r

t i t an ium, this could have been de te rmined b y previous

cons idera t ions ; however , the site posi t ion of the four

equ iva len t oxygen a toms appears to leave one the choices of e i ther C2h(4) or C2~(4) sites for, as no ted

before, b o t h sites will a c c o m m o d a t e four equ iva len t

atoms. Wyckof f ' s tables give the posi t ion of the oxygen a toms ou an (e) site. N o t i n g the a lphabet ic

t abu la t i on of sites above shows t h a t the (e) site is C2v. Therefore , there is no a m b i g u i t y in the choice of

site posi t ion for a toms, molecules, or ions if all t h e

i n fo rmat ion given in c rys ta l lographic tables is p rope r |y used.

A n o t h e r example of the p roper use of the crysta l lographic in fo rma t ion is the a-A120a crys ta l which is

R~c=--D:~ ~, Z = 2 . Consu l t ing Append ix I for D3d 6, we

find D3(2), C3~(2), C3(4), C~(6), C2(6), C1(12). Ob-


viously the four equivalent aluminum atoms can only be accommodated on the C~(4) site. However, the six equivalent oxygen atoms might be located in either the C~(6) or C2(6) site. Wyekoff 's table gives the following information for c~-A1203:

Site

A1 (c) Oxy (e).

Therefore, using the tables in Appendix I with the sites arranged in alphabetical order proceeding from left. to right, the following tabulat ion is implied.

Site in Alphabetical Appendix I" order Atom (site)

D3(2) a C~i(2) b C3(4) c Aluminum (c) Cd6) d C~(6) e Oxygen (e) C~(12) f

These two examples illustrate the point tha t it is a simple mat te r to determine the site symmet ry of an atom, molecule, or ion in a crystal lattice from information provided by x-ray experiments. Again, it is to be noted tha t all the sites are arranged in alphabetical order in Appendix I. The table in Appendix I is similar to tha t provided by Adams, 9 except for two major changes : (1) A reduction in the number of atoms found in the Bravais cell site from tha t given for the sites of the crystallographic cell; and (2) the arrange- ment of the site in alphabetical order to be consistent with the crystallographic tables.

APPENDIX III

The Method Suggested by Bhagavantan and Ven- katarayudu for Obtaining the Irreducible Represen- tation of a Crystal

Since SrTi03 has previously been t reated by the correlation method, it would be worth repeating this calculation using the method proposed by Bhagavan- tan and Venkatarayudu 3 to see (1) if both these methods give the same result and (2) demonstrate the simplicity of the correlation method. Only a simple outline of the Bhagavantan and Venkatarayudu is given below. (For a complete analysis, see Ref. 3.)

The irreducible representat ion for this crystal can be obtained as follows. By definition

wa = number of atoms left invariant under operation R.

x p = t h e character of the operation R, obtained in the following manner.

xp = ~0R (::t= 1-1- 2 cos0)

O

O F1G. 2. Unit cell of SrTiO~. . = T i Sr = strontium.

O

atom; O =0 atom ;

The angle 0 is defined as follows"

(1) where E is a proper rotation, 0 = 0 ° (2) (-t-) used for proper rotations, Cp, O= 360/P (3) ( - ) used for improper rotations, Sp, 0 = 360/P (4) ¢h is an improper rota t ion with 0 = 0 ° (5) i is an improper rotat ion with 0 = 180 °.

Next, each individual operation is considered in obtaining OOR and X~. Crystallographic information for SrTiO~, a perovskite, is 0h t - Pm3m. First the crystal s t ructure with atoms in the position of the unit cell must be considered. This same unit cell shown below will be used through this discussion. I t has been prev!ously noted tha t SrTiO~, a perovskite, is 012--Pm3m. A unit cell of this crystal s t ructure is shown in Fig. 2.

Note : (1) Each Ti is shared by 8 St; (2) There are 12 oxygens around 1 St.

To cheek this:

A lore co R

T i - -8 per cell, each contributing ~ to the unit cell . ' . 8 X ~ = 1 T i

Oxygen--12 atoms, each oxygen contr ibuted ¼ to unit cell . ' . 12X~= 3 0

S ~ 1 atom in middle of cell .'. 1 S r - - a t o m per unit cell 1 Sr

Tota l ooR =

for Z = 1

SrTiOz

(1) E Operation. Character and number of atoms invariant under E operation can be found as follows. All the atoms remained unchanged, .'.o OR=5 (i.e., 1 S r + l T i + 3 O). Xp=5(+1 - t -2 cos0°) = 5 . 3 = 15.

(2) C3 Operation. Figure 3(a) below shows the ¢'~ axes passing through this unit cell. The following list gives a tabulat ion of the number of atoms left in- var iant under all the C~ operations and the irreducible representat ion xp.


There are 7C~'s passing through the unit shown in Fig. 3(a); all are parallel to each other (this is not the 8C~ operation which appears in the character table, but only one of these eight operations).

Operation No. of atoms invariant, oo~

'C~ 1 S r + 2 . ~ Ti ~C~ ~ Ti ~C~ ~ Ti ~C~ ~ Ti ~C~ ~ Ti ¢C~ ~ Ti ~C~ ~ Ti

.'. 2koe = 2 Total atoms 1 S r + l Tr

However, for all C~ operations 1 + 2 c o s 0 (where 0= 120) = 0 .'.OaR(I--I-2 COS0) =0.

Since the above example illustrates the procedure followed on each symmet ry operation, only the basic essentials are presented in the discussion below for obtaining ~a and x~ for each operation.

Figure 3 (b).

Operation No. of atoms invariant, WR

• . ~ o 2 a = 3

'C~ 1 S r + 2 . ¼ 0 ~C~ ¼ 0 "C~ ~ O 4C~ 2. ~ Ti ~C~ ~ Ti ~C~ ~ Ti ~C~ 2- ~ Ti 9Ce ~ Ti I0C2 ~ Ti

Total a t o m s = l S r + l O + 1 Ti

Figure 3 ( e ) .

Operation ~e

'C4 1 Sr ~C~ 2.~ T i + ~ O ~C~ 2-~ T i + ¼ 0 4C~ 2.~ T i + ~ O ~C~ 2.~ T i + ~ O

.'. E w a = l S r + l T i + l 0 = 4

Figure 3 (d).

Operation ¢o~

1C2' 1 Sr 2C2' 2-~ T i + ~ O ~C2' 2. ~ O 4Cz' 2.~ T i + ~ O ~Cz' 2.41- O 6C~' 2.~ T i + ~ O ~C~' 2 . ¼ 0 ~C~' 2"~ T i + ¼ 0 "C~' 2- ¼ 0

.'. ~ = S r + T i - / - 3 0 = 5

There is a center of symmet ry at each and every atom in the unit ; .'. all the atoms remain invariant under one of the many /-inversion operations, i.e., 1 S r + l T i + 3 O, for Z~R=5.

The $4 operation yields the same result as the C4 operation even though there is the additional reflection. If we note tha t there are three reflection planes in the unit cell and all the atoms lie on one of these reflection planes and the $4 axis, .'. WR = 3. See Fig. 3 (e)

Figure 3 (f).

Operation WR

1S6 ~Ss 3S 6 ~S6 ~$6 6S 6 ~$6

• . ~ W R = 2

I S r + 2 . ~ Ti

Ti -~ Ti

Ti Ti Ti

In Fig. 3 (g) all atoms are invariant under z .'. ~ R = 5.

Figure 3 (h).

Operation (O~R

lo'd 2.~ T i + ~ O 2O~d 4-~ T i + 2 . ~ O + S r 3ad 2-~ T i + ~ O

• . ~ R = 3 ( T i + S r + O )

The results from above can be summarized in the following table :

Species W R X ~

(Oh factor group) E 5 15 8C3 2 0 6C2 3 - 3 6C4 3 3 3C~' 5 - 5 i 5 - 1 5 6S4 3 - 3 8S~ 2 0 3ah 5 5 6ae 3 3

Calculation of the number of modes in each of the species•

n(~) = number of modes in species g = order of the group g~ = number of elements in each species x~(~) = t h e character for the species i and irreducible

representat ion F ~ Xp~ =cha rac t e r of the irreducible representation, given above


6 / s

, /

°4 I I c 4 • i I

(c)

4C 2 sc~~%z

%

(b)

4S~ 2S~ ['IS~ I I ,

Planes ~ 7 ~

~ ~ . . . - l e~¢%~ ~ /Ldniersec?ion 7-" I ~ ' ~ I " ~ ~ ~ 0 - - ~ - - - - ~ - - ~ ~ of" Reflection I ....:.:..-. I

\ ~ 1 / i ts I I ® I I o6 (f) oz (g)

" ~ ' ,~ ~---o~.-,-- J J

(e)

!::1 (h)

x:::. "~1

Fro. 3. (a) Unit cell of SrTiO3 showing 7C3 operations ; (b) unit cell of SiTrO3 showing C~ operations ; (c) unit cell of SrTiOa showing C4 operations ; (d) unit cell of SrTiOa showing C2' operations ; (e) unit cell of SrTiOa showing $4 operations ; (f) unit cell of SrTiO3 showing S~ operations ; (g) unit cell of SrTiO~ showing the aa operations ; (h) unit cell of SrTiO~ showing the o-a operations.

E x a m p l e s of i t s u s e :

(1) A lg spec ies

Oh E 8C3 6C~ 6C4 3C2' X,A'~ = 1 1 1 1 1 xpi =15 0 --3 3 --5 gi = 1 8 6 6 3

i 6S4 8S6 3ah 6ad

1 1 1 ] 1 - 1 5 - 3 0 5 3

1 6 8 3 6

2;xiA~'xPi" g; = 15 +0 -- 18 + 18 -- 15 -- 15 -- 18 + 0 + 15 + 18 = O. g =48

• ". T h e r e are no l a t t i c e v i b r a t i o n s in A ~ species.

(2) F ~ spec ies

E 8C3 6C~ 6C4 3C~' i 6S4 8S6 3~,~ 6O'd

Xi Flu= 3 0 --1 1 --1 --3 --1 1 1 1 xp~ =15 0 --3 3 --5 --15 --3 0 5 3 g~ = 1 8 6 6 3 1 6 8 3 6

~x~x~i .g~ = 4 5 + 0 + 1 8 + 1 8 + 1 5 + 4 5 + 1 8 + 0 + 1 5 + 1 8 =4. g =48

• ". T h e r e a r e f o u r F ~ , i r r e d u c i b l e r e p r e s e n t a t i o n s .

(3) F 2 , spec ies

E 8C3 6C2 6C4 3C~' i 6S4 8S6 3~ . 6~ d

x f ~ . = 3 0 +1 --1 --1 --3 1 0 1 --1 x,~ =15 0 --3 3 --5 --15 --3 0 5 3 gl = 1 8 6 6 3 1 6 8 3 6

EX¢Xp~.g, :45+0 -- 18 -- 18+15+45 -- 1 8 + 0 + 1 5 -- 18 = 1. g=48

.'. T h e r e is 1 F2~ i r r educ ib le r e p r e s e n t a t i o n .

(4) A l l t h e o t h e r spec ies of 0h g i v e ze ro i r r e d u c i b l e

r e p r e s e n t a t i o n s .

(5) S u m m a r i z e p c r y s t a l = 4 F l , + F 2 , ~crystal __ pcrys ta l Facoustical 4 F ± F

n o w Xvibratio n - - x - - ~ l a T 2u - - t ' lu pcrystal __9~ .

vibration- or' l u-P ~' 2u.

T h i s c h e c k s o u r p r e v i o u s r e su l t .

I t is n o w e a s y to c o m p a r e t h e t w o m e t h o d s a n d see

t h a t t h e c o r r e l a t i o n m e t h o d t a k e s o n l y m i n u t e s wh i l e

t h i s m e t h o d i n v o l v e s h o u r s of work .

A P P E N D I X IV

C o r r e l a t i o n T a b l e s

C o r r e l a t i o n t a b l e s a r e a v a i l a b l e in m o s t b o o k s on

spectroscopy1°.11; h o w e v e r , we c h o o s e to s h o w t h e

d e r i v a t i o n of s e v e r a l t ab l e s . F o l l o w i n g t h e s e e x a m p l e s

o n e c a n a l l e v i a t e t h e p r o b l e m of c h o o s i n g t h e c o r r e c t

c o r r e l a t i o n t a b l e w h e n t w o o r m o r e pos s ib i l i t i e s ex is t ,

or , i n s o m e cases, w h e r e no d i r e c t c o r r e l a t i o n is g i v e n

in t h e p u b l i s h e d t ab l e s .

T h e f i rs t e x a m p l e is t h e s i m p l e c o r r e l a t i o n of t h e

p o i n t g r o u p C3, t o D3~. F i r s t , we w r i t e t h e p o i n t

g r o u p C ~ as fo l lows.

APPLIED SPECTROSCOPY 1'71

Car [ 2Ca (z) 3try

A~ 1 1 1 A2 1 1 - 1 E 2 --1 0

~'OW Car 1S a subgroup of Dab. This is easy to see for Da~ contains the same symmet ry operators as Ca,, i.e., I, 2Ca(z), 3 ~ plus additional operations ah, 2Sa, and 3~ . To obtain which species of Caw correlate with those species of Da~, one need only compare the character of those operations common to both point groups Da~ and Ca,; which in this case are I , 2Ca(z), 3~. To do this, we simply write the partial character table of Dab and including only the operations common to both Ca,. and Da~ as follows.

Operation Dab Caw Poini~

species I 2Ca (z) 3a~ group species

An' 1 1 1 A~ A~" 1 1 1"~ A~' 1 1 -- ~J A~ A~" 1 1 1 Ai E ' 2 --1 0~ E E " 2 -- 1

The species of each point group are compared, below, to find the correlation.

Operation

I 2Ca (z) 3a~

Point. group Da~ : species A~' 1 1 1 Point group Cat. : species A 1 1 1 1

Therefore, the correlation is Ca~A1 to Da~Alt. Also,

Operation

I 2C3(z) 3~r~

Point group Da~: species A{ ' 1 1 - 1 Point group Ca,: species A~ 1 1 - 1

The correlation is C~.A: to DahA1H.

The other correlations found as above are summarized below.

Dan Car Da~ Car

Ai ' A~ A~" A1 Aa" A2 E' E A~' A~. E" E

The correlation between D.,a and DAb is a bit more complicated since D4a contains two different subgroups which are identical to D2e. I t is best to illustrate this case with the specific example, first the point group of D~d can be wri t ten as follows.

Point group D2a I 2S4 (z) $ 4 ~ C ~ '' 2C~' 2ae

A1 1 1 1 1 1 A2 1 1 1 - 1 - 1 Bi 1 - 1 1 1 - 1 B2 1 --1 1 - 1 1 E 2 0 --2 0 0

Now the operations of D4~ which are similar to D2d a re

two sets: (1) I, 2S4(z), C~, 2C2', 2ze and (2) I, 2S4(z), C2, 2C2", 2~,. These subgroups differ only in the presence of the CJ and zd in subgroup 1 and the replacement of these operations with C~" and c% in subgroup 2. Repeat ing the procedure previously discussed, the operations common to both the point group Dth and D2a can be wri t ten as follows.

Species with same character

Subgroup 2 in point of D4h I 2S4(z) $42~C2 ', 2C2' 2ad group D2d

Alo 1 1 1 1 1 A~ A~u 1 - 1 1 1 - 1 B~ A2o 1 1 1 - 1 - 1 A2 A=u 1 - 1 1 - 1 1 B2 B~g 1 - 1 1 1 - 1 B~ B~, 1 1 1 1 1 A~ B2o 1 - 1 1 - 1 1 B2 B2. 1 1 1 --1 --1 A~ Eo 2 0 --2 0 0 E E~, 2 0 --2 0 0 E

Species with same character

Subgroup 2 in point of D4h I 284(z) $42-~C2 2C2" 2av group D2e

Alo 1 1 1 1 1 A i Alu 1 - -1 1 1 - 1 B1 Ago 1 1 1 - 1 - 1 A2 A ~ 1 --1 1 - 1 1 B2 Blo 1 --1 1 --1 1 B2 Blu 1 1 1 --1 --1 A2 B2~ 1 - 1 1 1 - 1 B~ B=. 1 1 1 1 1 A1 E~ 2 0 --2 0 0 E E~, 2 0 - 2 0 0 E

Summarizing :

(1) (2) C2' ~ C~' C~" ---~ C~'

D4h Deal D2d

Alo Ax A1 Ai,, B~ B~ A~ o A~ A2 A~,, B~ B2 Blo BI B2 Bi,, Ai A~ B2o B~ B1 B~, A~ A Eo E E E , E E

Therefore, to choose the correct correlation table one must consider the symmet ry elements in the site group and which of these symmet ry elements coincide

172. Volume 25, Number 2, 1971

with in the factor group. This is exactly the problem one faces in selecting the proper correlation tables in

the Ti02 example previously discussed. Here one must decide if the C~' element of the site D2e coincides with

the C2' element of the factor group. Also, the ~d plane of the site group must coincide with the ere plane of

the factor group, then one can choose the correlation tables marked C2'---~ C2'. However, if one finds the

C2' element coincides with the C~" operation of the

factor group and the ~ plane of site then coincides

with the ~d plane of the factor group then one must

choose the correlation tables where C2" ~ C2' as one does in the case of the t i tanium atom site in Ti02

and NH4I crystal. Tables relating the correct correla-

tion to be used for each site are available by writing to the authors. These tables eliminate the necessity

of relating the site symmet ry elements to the factor group symmetry, as described above.

1. D. F. Horn ig, J. Chem. Phys. 15, 1063 (1948). 2. H. Winston and R. S. Halford, J. Chem. Phys. 17, 607 (1949). 3. S. Bhagavantam and T. Venkatarayudu, Theory of Groups

and Its Application to Physical Problems (Bangalore Press, Bangalore City, India, 1951), 2nd ed.

4. C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966), 3rd ed.

5. J. C. Slater, Quantum Theory of Molecules and Solids (McGraw-Hill, New York, 1963-1967), Vols. I-III .

6. J. Durig and D. J. Antion, J. Chem. Phys. 51, 3639 (1969). 7. International Tables for X-Ray Crystallography, edited by

N. F. M. Henry and K. Lonsdale (T. Kynoch Press, Birming- ham, England, 1965), 2nd ed., Vol. 1.

8. R. W. C. Wyckoff, Crystal Structures (Interscience, New York, 1963, 1964), Vols. I and II.

9. E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations (McGraw-Hill, New York, 1955).

10. D. M. Adams, Metal-Ligand and Related Vibrations (St. Martin's Press, New York, 1968).

11. R. K. Khanna and C. W. Reimann, (Correlation Method), Spectra-Physics Raman Tech. Bull., No. 3, 1970.

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infrared and raman selection rules for lattice vibrations: the correlation method

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